Monday, June 16, 2014

Universal enveloping algebra

For a complex Lie algebra 𝔤, we can define its universal enveloping algebra U(𝔤) as the complex associative algebra T(𝔤)/I where T(𝔤) is the tensor algebra over 𝔤 and I is the ideal generated by {XYYX[X,Y]:X,Y𝔤}. We have the natural map ι:𝔤U(𝔤). The universal enveloping algebra satisfies the following universal property:

For any complex associative algebra A and linear map f:𝔤A satisfying [f(X),f(Y)]=f(X)f(Y)f(Y)f(X) for all X,Y𝔤, there is a unique algebra homomorphism f:U(𝔤)A such that f=fι.

Theorem (Poincare-Birkhoff-Witt). Let {X1,,Xm} be a basis for 𝔤. Then {(ιX1)n1(ιXm)nm:n1,,nm0} is a basis for U(𝔤).

Thursday, January 23, 2014

Reading note: A bounded operator approach to Tomita-Takesaki Theory (Rieffel, van Daele)

Let $\mathcal{H}$ be a real Hilbert space. We consider pairs $(\mathcal{K}, \mathcal{L})$ of closed subspaces of $\mathcal{H}$ satisfying the following properties: $\mathcal{K} \cap \mathcal{L} = \{0\}$ and $\mathcal{K} + \mathcal{L}$ is dense in $\mathcal{H}$.

...

Thursday, October 3, 2013

A list of boolean functions

Address function: $\textrm{Addr}_k: \{\pm 1\}^{k + 2^k} \rightarrow \{\pm 1\}$ is defined by \[ \textrm{Addr}_k(x_1, \ldots, x_k, y_1, \ldots, y_{2^k}) = y_x \] where $x = (x_1, \ldots, x_k)$ is interpreted as a number in $[2^k]$.